BAGHDAD, Iraq (CNN) -- The U.S. military will transfer control of Sunni Awakening groups to the Iraqi government next month, a U.S. military spokesman said Thursday in Baghdad.

An awakening council member in the Azamiyah area of Baghdad, Iraq, on Wednesday.

Also known as the Sons of Iraq, awakening councils are primarily a Sunni movement composed of former insurgents or sympathizers who have turned against al Qaeda.

The U.S. military credits them for having played a key role in bringing about the nationwide drop in violence that coincided with the "surge" in U.S. forces.

Members of the Awakening movement, targeted frequently by al Qaeda in Iraq in recent months, have complained that they receive little or no support from the Shiite-dominated government. Most of these groups' members want to be integrated into the Iraqi Security Forces, but few have been.

"The government of Iraq and coalition forces have agreed in principle to transfer all 100,000 Sons of Iraq," Maj. John Hall said. "The transfer will start with the Baghdad province, with the other provinces following at a later date."

According to the U.S. military, Baghdad province has contracted with approximately 54,000 of the Awakening members. The responsibility for their contracts will be transferred to the Iraqi government October 1, with the first payment from the government of Iraq to be made November 1, Hall said.

Coalition spokesman Brig. Gen. David Perkins said Monday that the timetable of the process will depend on how it goes in Baghdad.

"Based upon how that one works and the success we get from that, we will then start looking countrywide ... as we assimilate them into Iraqi society, as we get them jobs in the Iraqi security forces, within the ministries and into the civilian sector," he said.

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On Sunday, Iraqi government spokesman Ali al-Dabbagh told CNN's Arwa Damon that his government will pay Awakening members for a year.

The U.S. military said it pays the Baghdad province members' average monthly cost of $15 million, or $278 each.

In a move that has heightened tensions with the Awakening movement, the government recently issued a list bearing the names of Awakening leaders it wants detained.

On Wednesday, dozens of members of a local Awakening group and residents in the predominantly Sunni Adhamiya district of Baghdad took to the streets to protest the arrest list, threatening to act if the government cracks down. Adhamiya was once a stronghold of al Qaeda in Iraq, but over the past year, it has come under the control of Sunni fighters.

Al-Dabbagh responded to a question about what the government fears from the councils.

"The Iraqi government believes there are some undisciplined elements, some elements that are still connected to terrorists, and they blemish the Awakening, and they cannot be accepted as part of the Awakening," he said Sunday.

He credited the council for having "done a good job and made a big effort to impose security and fight al Qaeda and terrorist groups."

But, he added, "the Iraqi government has the right to filter individuals and to carry out thorough security checks."

The councils "cannot remain as armed groups uncontrolled by the government," al-Dabbagh said, but he added that it is up to the government "to find good job and life opportunities for those in the Awakening as much as it possibly can."

He described the concept of the Awakening movement as "a good and successful one" but said it has to change after the country's security situation stabilizes.

"They should take another form to contribute to building and serving the country," he said.